The Load Balancing Algorithm for the Star
Interconnection Network
Ahmad M. Awwad, Jehad Al-Sadi

Abstract—The star network is one of the promising
interconnection networks for future high speed parallel computers, it
is expected to be one of the future-generation networks. The star
network is both edge and vertex symmetry, it was shown to have
many gorgeous topological proprieties also it is owns hierarchical
structure framework. Although much of the research work has been
done on this promising network in literature, it still suffers from
having enough algorithms for load balancing problem. In this paper
we try to work on this issue by investigating and proposing an
efficient algorithm for load balancing problem for the star network.
The proposed algorithm is called Star Clustered Dimension Exchange
Method SCDEM to be implemented on the star network. The
proposed algorithm is based on the Clustered Dimension Exchange
Method (CDEM). The SCDEM algorithm is shown to be efficient in
redistributing the load balancing as evenly as possible among all
nodes of different factor networks
Keywords— Interconnection networks, Load balancing, Star
network.
I. INTRODUCTION
T
HE star graph was proposed by Akers and et al as one
of the attractive topologies, it was proposed as an
alternative to the cube network [1]. The star graph shown to
have excellent topological properties for comparable network
sizes of the cube network [2]. The star graph showed to have
many good properties over many networks including the wellknown cube network including: smaller diameter, smaller
degree, and smaller average diameter. The star graph proved
to have a hierarchical structure which will enable it building
large network size of smaller ones, the star graph is both edge
and vertex symmetry [1].
Although some algorithms proposed for the star graph such
as distributed fault-tolerant routing algorithm [3]. The
proposed algorithm adapts the routing decisions in response to
node failures, also the star network is shown to have fault
tolerance properties [3]. Anyway one of the main problems
that the star network still suffers from is having enough
algorithms for load balancing problem.
To our knowledge there is no enough results proposed in
literature about implementing and proposing efficient
algorithms for load balancing on star network. In this paper we
try to fill this gap by proposing and embedding the SCDEM
algorithm on the star graph which is based on the CDEM
Dr. Awwad works at University of Petra (awwad@uop.edu.jo),
Sadi works at Arab Open University (j_alsadi@aou.edu.jo)
Dr. Al-
algorithm which was shown to be attractive on OTISHypercube network by redistributing the load as evenly
among different processors [4]. Efficient implementation of
the SCDEM algorithm on the star network will make the star
network more acceptable network for real life application in
connection to load balancing problem. The rest of the paper is
organized as follows: In section 2 we present the necessary
basic notations and definitions, in section 3 we introduce some
of the related work on load balancing, in section 4 we present
and discuss the implementation of the SCDEM algorithm on
the star graph, also we present an example of SCDEM on S4
star network, finally section 5 concludes this paper.
II. DEFINITIONS AND BASIC TOPOLOGICAL PROPERTIES
During the last decade a huge number of interconnection
networks for High Speed Parallel Computers (HSPC) have
been investigated and proposed in literature [5] - [7]. As an
example one of these networks was the hypercube
interconnection network, also this network is known as the
binary n-cube. The star graph [1] is another example, which
has been proposed as an attractive alternative to the hypercube
network. Since its appearance the star network has attracted a
lot of research efforts. Some properties of this network have
been studied in the literature including its basic topological
properties [1], parallel path classification [4], node
connectivity [2] and embedding [8]. The authors Akers and
Krishnamurthy [3] have proved that the star graph has several
advantages over the hypercube network including a lower
degree for a fixed network size of the comparable network
sizes, a smaller diameter, and smaller average diameter.
Furthermore they showed that the star graph is maximally
fault tolerant edge, and vertex symmetric [3].
The structure of the star network can play an effective step
for proposing any algorithms on it. The authors in [21] Menn
and Somani have shown that the star graph may be seen as n 
(n-1)!, where the rows and the columns in this framework are
(n-1)-star and an n-linear array correspondingly. Also, Ferreria
and Berthome [9] claimed that the star graph may be seen also
as a rectangular framework RC (Rows by Columns) where
the rows are substar-Sn-2 and the columns are n (n-1) nodes on
each of its column.
However, there has been relatively a limited research efforts
have been dedicated to design efficient algorithms for the star
graph including computing fast Fourier transforms [10],
broadcasting [11], selection and sorting [12], [21], and Matrix
Multiplications [13] and load balancing [14], [15]. In an
attempt to overcome this problem we present an efficient
algorithm for load balancing problem on star graph to
redistribute the load balancing among all processors of the
network as evenly as possible.
Definition 1: The n-star graph, which is denoted by Sn, has
n nodes each labelled with a sole permutation n = {1,…,n}.
Any two nodes of Sn are connected if, and only if, their
corresponding permutations differ exactly in the first position
and any other position.
Fig. 1 shows the 4-star graph with 4 groups each containing
6 vertices (i.e. four copies of 3-star graphs). The degree, ,
and the diameter, , of the star graph are as follows [1], [16]:
, of the n-star graph = n-1, where n1.
the proposed algorithms on load balancing has shown a
considerably major advancement in enhancement of efficiency
and stability [18], [19]. In another research done by Zaho and
Xiao they have presented different DED-X schemes for load
balancing on homogeneous OTIS networks and they proposed
new algorithm structure called Generalized DiffusionExchange- Diffusion Method, the proposed scheme enabled
load balancing on Heterogeneous OTIS networks [20].
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III. BACKGROUND AND RELATED WORK
The attractive results shown and proved by researchers in
literature of the star graph make it a one of the strongest
competitor’s topology for High Speed Parallel Computers
(HSPC) and a strong candidate network for real life
applications. This fact has motivated us to investigate the load
balancing problem on the star network since the star graph
suffers from limited number of efficient algorithms proposed
for it in general and load balancing problem as specific case.
The load balancing problem has been investigated on various
types of infrastructure ranging from electronic networks [15]
and OTIS networks [4].
Load balancing problem is one of the well-known and
important types of problems which was studied from different
point views and different approaches. This problem was
studied and investigated by Ranka, Won, and Sahni [17], they
proposed and introduced the Dimension Exchange Method
(DEM) on the hypercube topology. The DEM algorithm was
based on the idea of finding the average load of neighbours’
nodes, such that the dimension of the network is n, the
neighbours which are connected on the nth dimension they will
exchange their loads to redistribute the load and achieve
evenly load balancing as possible, the processor which have
more load will broadcast the extra amount of the load to its
direct neighbour node. The main advantage of the Dimension
Exchange Method is that every node will be able to
redistribute tasks to its direct neighbours to reach even load
balancing among all nodes. Ranka and et al have achieved that
in the worst case in the DEM method to redistribute load
balancing was log2n on the cube network [17].
The researchers Zaho, Xiao, and Qin have presented hybrid
scheme of diffusion and dimension exchange called DED-X
for load balancing on Optical Transpose Interconnection
System (OTIS) [18] , [19]. The proposed algorithm works by
dividing the load balancing task to three different phases. The
results achieved on OTIS networks showed that the load
balance efficiently redistributed almost evenly. On the other
hand the achieved results of the simulation from Zaho et al of
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Fig. 1: The 4-Star Graph
Furthermore Zaho, Xiao, and Qin have shown that the
usability of the new proposed load balancing methods to be
better than the X traditional load balancing algorithm [20].
The main objective of this paper is to propose and present a
new load balancing algorithm for the star networks named Star
Clustered Dimension Exchange Method (SCDEM) based on
the algorithm [4].
The Star Clustered Dimension Exchange Method for Load
Balancing on the Star Network
The algorithm we present in this paper SCDEM is based on
the Clustered Dimension Exchange Method CDEM for load
balancing for Optical Transpose Interconnection system on
Hypercube factor network [4]. The worst time case complexity
of CDEM for load balancing on OTIS-Hypercube was
O(Sqrt(p)*M log2 p). Also the number of communication
steps which is required by CDEM proved to be 3log2 p [4].
The main achievement of the new presented SCDEM is to
obtain even load balancing for the Sn network by
redistributing number of tasks between different nodes on
different groups. The number cooperating moves needed
between different nodes in the SDEM is 2n-1, where n is the
degree of Sn. Fig. 2 presents the SCDEM algorithm for load
balancing problem on n! Processors of Sn.
same process will be repeated continually until it reach the
The SCDEM load balancing algorithm is based on the
following 3 phases:
٠Phase 1: The load balancing of neighbour nodes of the 1st
stage is achieved by redistributing the load balancing of all
direct neighbour nodes, and only if, their corresponding
permutations differ exactly in the 1st and 2nd position.
Then redistributing the load balancing of any two neighbour
nodes that are connected if, and only if, their corresponding
permutations differ exactly in the first and 3rd position.
Keep redistributing the load balancing of any two neighbour
nodes “as above” that are connected if, and only if, their
corresponding permutations differ exactly in the first and nth
position.
٠Phase 2: Repeat phase one more time.
٠Phase 3: For each direct neighbour’s processors pi and pj
find the maximum difference weight of the directed weights
and redistribute the weight among the nodes of highest
difference following the steps 3-12.
Note that n-1 is the number of neighbours of any
processor in Sn:
neighbours pj that is n positions far away from pi.
٠Phase 2: To enhance the load balancing efficiency between
different processors of n factor networks, the algorithm
suggests repeating the steps 2 to 12 as mentioned above.
٠Phase 3: As a final phase all adjacent processors which they
differ in first position and any other position i.e. pi and pj. The
algorithms will find the maximum difference among all the
weights of these neighbors and redistribute the weight
between pi and pj with max difference, │pi - pj│ following the
steps 2-12 of the SCDEM algorithm.
3.
4.
5.
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7.
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10.
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12.
13.
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for n = 1; n ≤ n-1; n++
for all neighbour nodes pi and pj which they differ in
1st and n+1 position of Sn do in parallel
Give-and-take pi and pj total load sizes of the two
nodes
TheAverageLoad pi,j = Floor (Load pi + Load pi)/2
if ( Totalload pi >= excess AverageLoad pi,j )
Send excess load pi to the neighbour node pi
Load pi = Load pi – extra load
Load pj = Load pj + extra load
else
Receive extra load from neighbour pj
Load pi = Load pi + extra load
Load pj = Load pj – extra load
Repeat steps 3 to 12 one more time
for all adjacent processors of pi and find the max │pi
– pj,│ such that pj is the set of all neighbours of pi
where 1≤ j ≤ n-1.
redistribute the weight as in steps as in steps 3 to12
between pi and the node with the max │pi – pj│.
Fig. 2: The SCDEM load balancing Algorithm
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Fig. 3: 4-star network – load balancing - initial state
Example: - To explain the SCDEM algorithm presented in
Fig. 2, the following example implements the load balancing
algorithm on the different factor networks of S4.
SCDEM algorithm works on redistributing load balancing
Fig. 3 shows the four factor networks 4-S3 of the Network
among all processors of the network, phases: one, two and
S4, each factor network has 6 processors with a specific load
three are done in parallel.
assigned to it. Since the degree of Sn is n-1 it follows that each
٠Phase 1: The load balancing between the processors of Sn
node connected to three other direct nodes, two of the inner
based on SCDEM algorithm is exchanged as in steps 2 to 12
in parallel, in first step the load exchange will be between all
the processors in which they differ in 1 st position and 2nd
position for all the factor networks of Sn i.e. Sn -1. Then the
group and one in outer group. The number which was assigned
next to the processor presents the starting load.
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of this phase Fig. 6 shows the new load balancing distribution
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of the phase 1 of the algorithm.
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Fig. 4: 4-Star network – load balancing phase 1-A
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Fig. 6: 4-Star network – load balancing phase 1-C
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following the steps 2-12. Fig.s 4, 5 and 6 reflects phase1: A, B
and C of the SCDEM algorithm, each pair of nodes which
they differ in 1st position, 2nd, 3rd and 4th position. At the end
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First we start by implanting phase 1 of the algorithm by
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Fig 5: 4-Star network – processors load balancing phase 1-B
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Fig.7: 4-Star network –load balancing – phase 2
In phase 2 we repeat the same steps one more time to
redistribute the load balancing among the neighbours’ nodes
as suggested in SCDEM algorithm presented in Fig. 2 by
following the steps 2- 12. Fig. 7 shows the load balancing of
As future extension of this research work we will do some
all processors at the end of phase 2.
analytical estimation including: execution time, load balancing
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accuracy, communication steps and speed to prove the
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SCDEM efficiency mathematically.
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[1]
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[2]
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Fig. 8: 4-Star network – load balancing – phase 3
Finally in phase 3 all adjacent nodes which differ in first
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[9]
[10]
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[11]
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[12]
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[13]
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[14]
network.
[15]
IV.
CONCLUSION
This paper presented an efficient algorithm for distributing
[16]
the load balancing of nodes in star network. The proposed
algorithm is called Star Clustered Dimension Exchange
[17]
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[18]
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